Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls

Dianetti J, Ferrari G (2021) Center for Mathematical Economics Working Papers; 645.
Bielefeld: Center for Mathematical Economics.

Diskussionspapier | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The considered optimization problems concern the minimization of a discounted cost functional over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. The setting is multidimensional, the dynamics of the state and the costs are convex, the volatility matrix can be constant or linear in the state. We prove that the optimal control acts only when the underlying diffusion attempts to exit the so-called waiting region, and that the direction of this action is prescribed by the derivative of the value function. Our approach is based on the study of a suitable monotonicity property of the derivative of the value function through its interpretation as the value of an optimal stopping game. Such a monotonicity allows to construct nearly optimal policies which reflect the underlying diffusion at the boundary of approximating waiting regions. The limit of this approximation scheme then provides the desired characterization. Our result applies to a relevant class of linear-quadratic models, among others. Furthermore, it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature, where this important aspect was only partially addressed.

AMS subject classification: 93E20, 60G17, 91A55, 49J40
Stichworte
Dynkin games; re ected diffusion; singular stochastic control; Skorokhod problem; variational inequalities
Erscheinungsjahr
2021
Band
645
Seite(n)
38
ISSN
0931-6558
Page URI
https://pub.uni-bielefeld.de/record/2952857

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Dianetti J, Ferrari G. Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls. Center for Mathematical Economics Working Papers. Vol 645. Bielefeld: Center for Mathematical Economics; 2021.
Dianetti, J., & Ferrari, G. (2021). Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls (Center for Mathematical Economics Working Papers, 645). Bielefeld: Center for Mathematical Economics.
Dianetti, J., and Ferrari, G. (2021). Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls. Center for Mathematical Economics Working Papers, 645, Bielefeld: Center for Mathematical Economics.
Dianetti, J., & Ferrari, G., 2021. Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls, Center for Mathematical Economics Working Papers, no.645, Bielefeld: Center for Mathematical Economics.
J. Dianetti and G. Ferrari, Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls, Center for Mathematical Economics Working Papers, vol. 645, Bielefeld: Center for Mathematical Economics, 2021.
Dianetti, J., Ferrari, G.: Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls. Center for Mathematical Economics Working Papers, 645. Center for Mathematical Economics, Bielefeld (2021).
Dianetti, Jodi, and Ferrari, Giorgio. Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls. Bielefeld: Center for Mathematical Economics, 2021. Center for Mathematical Economics Working Papers. 645.
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2021-03-18T08:30:22Z
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